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| {{Group theory sidebar |Basics}}
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| In [[group theory]], the '''direct product''' is an operation that takes two [[Group (mathematics)|groups]] {{math|''G''}} and {{math|''H''}} and constructs a new group, usually denoted {{math|''G'' × ''H''}}. This operation is the group-theoretic analogue of the [[Cartesian product]] of [[Set (mathematics)|set]]s and is one of several important notions of [[direct product]] in mathematics.
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| In the context of [[abelian group]]s, the direct product is sometimes referred to as the [[direct sum]], and is denoted {{math|''G'' ⊕ ''H''}}. Direct sums play an important role in the classification of abelian groups: according to the [[fundamental theorem of finite abelian groups]], every finite abelian group can be expressed as the direct sum of [[cyclic group]]s.
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| == Definition ==
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| Given groups {{math|''G''}} and {{math|''H''}}, the '''direct product''' {{math|''G'' × ''H''}} is defined as follows:
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| # The elements of {{math|''G'' × ''H''}} are [[ordered pair]]s {{math|(''g'', ''h'')}}, where {{math|''g'' ∈ ''G''}} and {{math|''h'' ∈ ''H''}}. That is, the set of elements of {{math|''G'' × ''H''}} is the Cartesian product of the sets {{math|''G''}} and {{math|''H''}}.
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| # The [[binary operation]] on {{math|''G'' × ''H''}} is defined componentwise: <center>{{math|(''g''<sub>1</sub>, ''h''<sub>1</sub>) · (''g''<sub>2</sub>, ''h''<sub>2</sub>) {{=}} (''g''<sub>1</sub> · ''g''<sub>2</sub>, ''h''<sub>1</sub> · ''h''<sub>2</sub>)}}</center>
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| The resulting algebraic object satisfies the axioms for a group. Specifically:
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| ;Associativity: The binary operation on {{math|''G'' × ''H''}} is indeed [[Associativity|associative]].
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| ;Identity: The direct product has an [[identity element]], namely {{math|(1<sub>''G''</sub>, 1<sub>''H''</sub>)}}, where {{math|1<sub>''G''</sub>}} is the identity element of {{math|''G''}} and {{math|1<sub>''H''</sub>}} is the identity element of {{math|''H''}}.
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| ;Inverses: The [[inverse element|inverse]] of an element {{math|(''g'', ''h'')}} of {{math|''G'' × ''H''}} is the pair {{math|(''g''<sup>−1</sup>, ''h''<sup>−1</sup>)}}, where {{math|''g''<sup>−1</sup>}} is the inverse of {{math|''g''}} in {{math|''G''}}, and {{math|''h''<sup>−1</sup>}} is the inverse of {{math|''h''}} in {{math|''H''}}.
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| ==Examples==
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| * Let {{math|'''R'''}} be the group of [[real number]]s under [[addition]]. Then the direct product {{math|'''R''' × '''R'''}} is the group of all two-component [[Euclidean vector|vectors]] {{math|(''x'', ''y'')}} under the operation of [[Euclidean vector#Addition and subtraction|vector addition]]: <center>{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>) + (''x''<sub>2</sub>, ''y''<sub>2</sub>) {{=}} (''x''<sub>1</sub> + ''x''<sub>2</sub>, ''y''<sub>1</sub> + ''y''<sub>2</sub>)}}.</center>
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| * Let {{math|''G''}} and {{math|''H''}} be [[cyclic group]]s with two elements each:
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| {|
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| | style="width:70px;" |
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| {| class="wikitable" style="text-align:center; width:90px; height:90px;"
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| |+ {{math|''G''}}
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| |-
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| ! * !! {{math|1}} !! {{math|''a''}}
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| |-
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| ! {{math|1}}
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| | {{math|1}} || {{math|''a''}}
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| |-
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| ! {{math|''a''}}
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| | {{math|''a''}} || {{math|1}}
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| |}
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| | style="width:50px;" |
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| {| class="wikitable" style="text-align:center; width:90px; height:90px;"
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| |+ {{math|''H''}}
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| |-
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| ! * !! {{math|1}} !! {{math|''b''}}
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| |-
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| ! {{math|1}}
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| | {{math|1}} || {{math|''b''}}
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| |-
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| ! {{math|''b''}}
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| | {{math|''b''}} || {{math|1}}
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| |}
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| |}
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| :Then the direct product {{math|''G'' × ''H''}} is [[group isomorphism|isomorphic]] to the [[Klein four-group]]:
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| :::{| class="wikitable" style="text-align:center; width:250px; height:175px;"
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| |+ {{math|''G'' × ''H''}}
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| |-
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| ! * !! {{math|(1, 1)}} !! {{math|(''a'', 1)}} !! {{math|(1, ''b'')}} !! {{math|(''a'', ''b'')}}
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| |-
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| ! {{math|(1, 1)}}
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| | {{math|(1, 1)}} || {{math|(''a'', 1)}} || {{math|(1, ''b'')}} || {{math|(''a'', ''b'')}}
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| |-
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| ! {{math|(''a'', 1)}}
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| | {{math|(''a'', 1)}} || {{math|(1, 1)}} || {{math|(''a'', ''b'')}} || {{math|(1, ''b'')}}
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| |-
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| ! {{math|(1, ''b'')}}
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| | {{math|(1, ''b'')}} || {{math|(''a'', ''b'')}} || {{math|(1, 1)}} || {{math|(''a'', 1)}}
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| |-
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| ! {{math|(''a'', ''b'')}}
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| | {{math|(''a'', ''b'')}} || {{math|(1, ''b'')}} || {{math|(''a'', 1)}} || {{math|(1, 1)}}
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| |}
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| ==Elementary properties==
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| * The [[Order (group theory)|order]] of a direct product {{math|''G'' × ''H''}} is the product of the orders of {{math|''G''}} and {{math|''H''}}:
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| ::<big>|</big>{{math| ''G'' × ''H'' }}<big>| = |</big>{{math| ''G'' }}<big>| |</big>{{math| ''H'' }}<big>|</big>.
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| :This follows from the formula for the [[cardinality]] of the cartesian product of sets.
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| * The order of each element {{math|(''g'', ''h'')}} is the [[least common multiple]] of the orders of {{math|''g''}} and {{math|''h''}}:<ref>{{Cite book|title = Contemporary Abstract Algebra|last = Gallian|first = Joseph A.|publisher = Cengage Learning|year = 2010|isbn = 9780547165097|location = |pages = |edition = 7|page = 157}}</ref>
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| ::<big>|</big>{{math| (''g'', ''h'') }}<big>| = </big>{{math|lcm}}<big>( |</big>{{math| ''g'' }}<big>|, |</big>{{math| ''h'' }}<big>| )</big>.
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| :In particular, if <big>|</big>{{math| ''g'' }}<big>|</big> and <big>|</big>{{math| ''h'' }}<big>|</big> are [[relatively prime]], then the order of {{math| (''g'', ''h'') }} is the product of the orders of {{math| ''g'' }} and {{math| ''h'' }}.
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| * As a consequence, if {{math|''G''}} and {{math|''H''}} are [[cyclic groups]] whose orders are relatively prime, then {{math|''G'' × ''H''}} is cyclic as well. That is, if {{math|''m''}} and {{math|''n''}} are relatively prime, then
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| ::{{math|( '''Z''' / ''m'''''Z''' ) × ( '''Z''' / ''n'''''Z''' ) <big>≅</big> '''Z''' / ''mn'''''Z'''}}.
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| :This fact is closely related to the [[Chinese remainder theorem]].
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| ==Algebraic structure==
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| Let {{math|''G''}} and {{math|''H''}} be groups, let {{math|''P'' {{=}} ''G'' × ''H''}}, and consider the following two [[subset]]s of {{math|''P''}}:
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| ::G' = <big>{</big> {{math|(''g'', 1) : ''g'' ∈ ''G''}} <big>}</big> and H' = <big>{</big> {{math|(1, ''h'') : ''h'' ∈ ''H''}} <big>}</big>
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| Both of these are in fact [[subgroup]]s of {{math|''P''}}, the first being isomorphic to {{math|''G''}}, and the second being isomorphic to {{math|''H''}}. If we identify these with {{math|''G''}} and {{math|''H''}}, respectively, then we can think of the direct product {{math|''P''}} as containing the original groups {{math|''G''}} and {{math|''H''}} as subgroups.
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| These subgroups of {{math|''P''}} have the following three important properties:
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| (Saying again that we identify G' and H' with G and H, respectively.)
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| # The [[Intersection (set theory)|intersection]] {{math|''G'' ∩ ''H''}} is [[Trivial group|trivial]].
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| # Every element of {{math|''P''}} can be expressed as the product of an element of {{math|''G''}} and an element of {{math|''H''}}.
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| # Every element of {{math|''G''}} [[commutativity|commutes]] with every element of {{math|''H''}}.
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| Together, these three properties completely determine the algebraic structure of the direct product {{math|''P''}}. That is, if {{math|''P''}} is ''any'' group having subgroups {{math|''G''}} and {{math|''H''}} that satisfy the properties above, then {{math|''P''}} is necessarily isomorphic to the direct product of {{math|''G''}} and {{math|''H''}}. In this situation, {{math|''P''}} is sometimes referred to as the '''internal direct product''' of its subgroups {{math|''G''}} and {{math|''H''}}.
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| In some contexts, the third property above is replaced by the following:
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| :3'. Both {{math|''G''}} and {{math|''H''}} are [[normal subgroup|normal]] in {{math|''P''}}.
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| This property is equivalent to property 3, since the elements of two normal subgroups with trivial intersection necessarily commute, a fact which can be deduced by considering the [[commutator]] [''g'',''h''] of any ''g'' in ''G'', ''h'' in ''H''.
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| ===Examples===
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| * Let {{math|''V''}} be the [[Klein four-group]]:
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| ::{| class="wikitable" style="text-align:center; width:150px; height:150px;"
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| |+ {{math|''V''}}
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| |-
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| ! * !! {{math|1}} !! {{math|''a''}} !! {{math|''b''}} !! {{math|''c''}}
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| |-
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| ! {{math|1}}
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| | {{math|1}} || {{math|''a''}} || {{math|''b''}} || {{math|''c''}}
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| |-
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| ! {{math|''a''}}
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| | {{math|''a''}} || {{math|1}} || {{math|''c''}} || {{math|''b''}}
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| |-
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| ! {{math|''b''}}
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| | {{math|''b''}} || {{math|''c''}} || {{math|1}} || {{math|''a''}}
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| |-
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| ! {{math|''c''}}
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| | {{math|''c''}} || {{math|''b''}} || {{math|''a''}} || {{math|1}}
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| |}
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| :Then {{math|''V''}} is the internal direct product of the two-element subgroups { {{math|1, ''a''}} } and { {{math|1, ''b''}} }.
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| * Let {{math|〈''a''〉}} be a cyclic group of order {{math|''mn''}}, where {{math|''m''}} and {{math|''n''}} are relatively prime. Then {{math|〈''a<sup>n</sup>''〉}} and {{math|〈''a<sup>m</sup>''〉}} are cyclic subgroups of orders {{math|''m''}} and {{math|''n''}}, respectively, and {{math|〈''a''〉}} is the internal direct product of these subgroups.
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| * Let {{math|'''C'''<sup>×</sup>}} be the group of nonzero [[complex number]]s under [[multiplication]]. Then {{math|'''C'''<sup>×</sup>}} is the internal direct product of the [[circle group]] {{math|'''T'''}} of unit complex numbers and the group {{math|'''R'''<sup>+</sup>}} of [[positive number|positive]] real numbers under multiplication.
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| * If ''n'' is odd, then the [[general linear group]] {{math|''GL''(''n'', '''R''')}} is the internal direct product of the [[special linear group]] {{math|''SL''(''n'', '''R''')}} and the subgroup consisting of all [[scalar matrix|scalar matrices]].
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| * Similarly, when ''n'' is odd the [[orthogonal group]] {{math|''O''(''n'', '''R''')}} is the internal direct product of the special orthogonal group {{math|''SO''(''n'', '''R''')}} and the two-element subgroup { {{math|−''I'', ''I''}} }, where {{math|''I''}} denotes the [[identity matrix]].
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| * The [[symmetry group]] of a [[cube]] is the internal direct product of the subgroup of rotations and the two-element group { {{math|−''I'', ''I''}} }, where {{math|''I''}} is the identity element and {{math|−''I''}} is the [[point reflection]] through the center of the cube. A similar fact holds true for the symmetry group of an [[icosahedron]].
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| * Let {{math|''n''}} be odd, and let {{math|''D''<sub>4''n''</sub>}} be the [[dihedral group]] of order {{math|4''n''}}:
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| ::{{math|''D''<sub>4''n''</sub>}} = {{math|〈 ''r'', ''s'' }}<big>|</big>{{math| ''r''<sup>2''n''</sup> {{=}} ''s''<sup>2</sup> {{=}} 1, ''sr'' {{=}} ''r''<sup>−1</sup>''s'' 〉}}.
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| :Then {{math|''D''<sub>4''n''</sub>}} is the internal direct product of the subgroup {{math|〈 ''r''<sup>2</sup>, ''s'' 〉}} (which is isomorphic to {{math|''D''<sub>2''n''</sub>}}) and the two-element subgroup { {{math|1, ''r''<sup>n</sup>}} }.
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| ===Presentations===
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| The algebraic structure of {{math|''G'' × ''H''}} can be used to give a [[Presentation of a group|presentation]] for the direct product in terms of the presentations of {{math|''G''}} and {{math|''H''}}. Specifically, suppose that
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| :{{math|''G''}} = {{math|〈 ''S<sub>G</sub>'' }}<big>|</big>{{math| ''R<sub>G</sub>'' 〉}} and {{math|''H''}} = {{math|〈 ''S<sub>H</sub>'' }}<big>|</big>{{math| ''R<sub>H</sub>'' 〉}},
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| where {{math|''S<sub>G</sub>''}} and {{math|''S<sub>H</sub>''}} are (disjoint) [[Generating set of a group|generating sets]] and {{math|''R<sub>G</sub>''}} and {{math|''R<sub>H</sub>''}} are defining relations. Then
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| :{{math|''G'' × ''H''}} = {{math|〈 ''S<sub>G</sub>'' ∪ ''S<sub>H</sub>'' }}<big>|</big>{{math| ''R<sub>G</sub>'' ∪ ''R<sub>H</sub>'' ∪ ''R<sub>P</sub>'' 〉}}
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| where {{math|''R<sub>P</sub>''}} is a set of relations specifying that each element of {{math|''S<sub>G</sub>''}} commutes with each element of {{math|''S<sub>H</sub>''}}
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| For example, suppose that
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| :{{math|''G''}} = {{math|〈 ''a'' }}<big>|</big>{{math| ''a''<sup>3</sup> {{=}} 1 〉}} and {{math|''H''}} = {{math|〈 ''b'' }}<big>|</big>{{math| ''b''<sup>5</sup> {{=}} 1 〉}}.
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| Then
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| :{{math|''G'' × ''H''}} = {{math|〈 ''a'', ''b'' }}<big>|</big>{{math| ''a''<sup>3</sup> {{=}} 1, ''b''<sup>5</sup> {{=}} 1, ''ab'' {{=}} ''ba'' 〉}}.
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| ===Normal structure===
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| As mentioned above, the subgroups {{math|''G''}} and {{math|''H''}} are normal in {{math|''G'' × ''H''}}. Specifically, define functions {{math|''π<sub>G</sub>'': ''G'' × ''H'' → ''G''}} and {{math|''π<sub>H</sub>'': ''G'' × ''H'' → ''H''}} by
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| :{{math|''π<sub>G</sub>''(''g'', ''h'') {{=}} ''g''}} and {{math|''π<sub>H</sub>''(''g'', ''h'') {{=}} ''h''}}.
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| Then {{math|''π<sub>G</sub>''}} and {{math|''π<sub>H</sub>''}} are [[group homomorphism|homomorphisms]], known as '''[[Projection (mathematics)|projection]] homomorphisms''', whose kernels are {{math|''H''}} and {{math|''G''}}, respectively.
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| It follows that {{math|''G'' × ''H''}} is an [[Group extension|extension]] of {{math|''G''}} by {{math|''H''}} (or vice-versa). In the case where {{math|''G'' × ''H''}} is a [[finite group]], it follows that the [[composition factor]]s of {{math|''G'' × ''H''}} are precisely the [[union (set theory)|union]] of the composition factors of {{math|''G''}} and the composition factors of {{math|''H''}}.
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| ==Further properties==
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| ===Universal property===
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| {{main | Product (category theory)}}
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| The direct product {{math|''G'' × ''H''}} can be characterized by the following [[universal property]]. Let {{math|''π<sub>G</sub>'': ''G'' × ''H'' → ''G''}} and {{math|''π<sub>H</sub>'': ''G'' × ''H'' → ''H''}} be the projection homomorphisms. Then for any group {{math|''P''}} and any homomorphisms {{math|ƒ<sub>''G''</sub>: ''P'' → ''G''}} and {{math|ƒ<sub>''H''</sub>: ''P'' → ''H''}}, there exists a unique homomorphism {{math|ƒ: ''P'' → ''G'' × ''H''}} making the following diagram [[Commutative diagram|commute]]:
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| :[[Image:DirectProductDiagram.png|200px]]
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| Specifically, the homomorphism {{math|ƒ}} is given by the formula
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| :{{math|ƒ(''p'') {{=}} <big>(</big> ƒ<sub>''G''</sub>(''p''), ƒ<sub>''H''</sub>(''p'') <big>)</big>}}.
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| This is a special case of the universal property for products in [[category theory]].
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| ===Subgroups===
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| If {{math|''A''}} is a subgroup of {{math|''G''}} and {{math|''B''}} is a subgroup of {{math|''H''}}, then the direct product {{math|''A'' × ''B''}} is a subgroup of {{math|''G'' × ''H''}}. For example, the isomorphic copy of {{math|''G''}} in {{math|''G'' × ''H''}} is the product {{math|''G'' × }}<big>{1}</big>, where <big>{1}</big> is the [[trivial group|trivial]] subgroup of {{math|''H''}}.
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| If {{math|''A''}} and {{math|''B''}} are normal, then {{math|''A'' × ''B''}} is a normal subgroup of {{math|''G'' × ''H''}}. Moreover, the [[Quotient group|quotient]] {{math|(''G'' × ''H'') / (''A'' × ''B'')}} is isomorphic to the direct product of the quotients {{math|''G'' / ''A''}} and {{math|''H'' / ''B''}}:
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| :{{math|(''G'' × ''H'') / (''A'' × ''B'')}} <big>≅</big> {{math|(''G'' / ''A'') × (''H'' / ''B'')}}. | |
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| Note that it is not true in general that every subgroup of {{math|''G'' × ''H''}} is the product of a subgroup of {{math|''G''}} with a subgroup of {{math|''H''}}. For example, if {{math|''G''}} is any group, then the product {{math|''G'' × ''G''}} has a [[diagonal subgroup]]
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| :{{math|Δ}} = { {{math|(''g'', ''g'') : ''g'' ∈ ''G''}} }
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| which is not the direct product of two subgroups of {{math|''G''}}. Other subgroups include fiber products of {{math|''G''}} and {{math|''H''}} (see below). The subgroups of direct products are described by [[Goursat's lemma]].
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| ===Conjugacy and centralizers===
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| Two elements {{math|(''g''<sub>1</sub>, ''h''<sub>1</sub>)}} and {{math|(''g''<sub>2</sub>, ''h''<sub>2</sub>)}} are [[conjugacy class|conjugate]] in {{math|''G'' × ''H''}} if and only if {{math|''g''<sub>1</sub>}} and {{math|''g''<sub>2</sub>}} are conjugate in {{math|''G''}} and {{math|''h''<sub>1</sub>}} and {{math|''h''<sub>2</sub>}} are conjugate in {{math|''H''}}. It follows that each conjugacy class in {{math|''G'' × ''H''}} is simply the Cartesian product of a conjugacy class in {{math|''G''}} and a conjugacy class in {{math|''H''}}.
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| Along the same lines, if {{math|(''g'', ''h'') ∈ ''G'' × ''H''}}, the [[centralizer]] of {{math|(''g'', ''h'')}} is simply the product of the centralizers of {{math|''g''}} and {{math|''h''}}:
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| :{{math|''C''<sub>''G''×''H''</sub>(''g'', ''h'')}} = {{math|''C''<sub>''G''</sub>(''g'') × ''C''<sub>''H''</sub>(''h'')}}.
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| Similarly, the [[Center (group theory)|center]] of {{math|''G'' × ''H''}} is the product of the centers of {{math|''G''}} and {{math||''H''}}:
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| :{{math|''Z''(''G'' × ''H'')}} = {{math|''Z''(''G'') × ''Z''(''H'')}}.
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| [[Normalizer]]s behave in a more complex manner since not all subgroups of direct products themselves decompose as direct products.
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| ===Automorphisms and endomorphisms=== | |
| If {{math|''α''}} is an [[group automorphism|automorphism]] of {{math|''G''}} and {{math|''β''}} is an automorphism of {{math|''H''}}, then the product function {{math|''α'' × ''β'': ''G'' × ''H'' → ''G'' × ''H''}} defined by
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| :{{math|(''α'' × ''β'')(''g'', ''h'') {{=}} <big>(</big>''α''(''g''), ''β''(''h'')<big>)</big>}}
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| is an automorphism of {{math|''G'' × ''H''}}. It follows that {{math|Aut(''G'' × ''H'')}} has a subgroup isomorphic
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| to the direct product {{math|Aut(''G'') × Aut(''H'')}}.
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| It is not true in general that every automorphism of {{math|''G'' × ''H''}} has the above form. (That is, {{math|Aut(''G'') × Aut(''H'')}} is often a proper subgroup of {{math|Aut(''G'' × ''H'')}}.) For example, if {{math|''G''}} is any group, then there exists an automorphism {{math|''σ''}} of {{math|''G'' × ''G''}} that switches the two factors, i.e.
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| :{{math|''σ''(''g''<sub>1</sub>, ''g''<sub>2</sub>) {{=}} (''g''<sub>2</sub>, ''g''<sub>1</sub>)}}.
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| For another example, the automorphism group of {{math|'''Z''' × '''Z'''}} is [[Modular group|{{math|''GL''(2, '''Z''')}}]], the group of all {{math|2 × 2}} [[Matrix (mathematics)|matrices]] with integer entries and [[determinant]] {{math|±1}}. This automorphism group is infinite, but only finitely many of the automorphisms have the form given above.
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| In general, every [[endomorphism]] of {{math|''G'' × ''H''}} can be written as a {{math|2 × 2}} matrix
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| :<math>\begin{bmatrix}\alpha & \beta \\ \gamma & \delta\end{bmatrix}</math>
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| where {{math|''α''}} is an endomorphism of {{math|''G''}}, {{math|''δ''}} is an endomorphism of {{math|''H''}}, and {{math| ''β'': ''H'' → ''G''}} and {{math| ''γ'': ''G'' → ''H''}} are homomorphisms. Such a matrix must have the property that every element in the [[Image (mathematics)|image]] of {{math|''α''}} commutes with every element in the image of {{math|''β''}}, and every element in the image of {{math|''γ''}} commutes with every element in the image of {{math|''δ''}}.
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| When ''G'' and ''H'' are indecomposable, centerless groups, then the automorphism group is relatively straightforward, being Aut(''G'') × Aut(''H'') if ''G'' and ''H'' are not isomorphic, and Aut(''G'') wr 2 if ''G'' ≅ ''H'', wr denotes the [[wreath product]]. This is part of the [[Krull–Schmidt theorem]], and holds more generally for finite direct products.
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| ==Generalizations==
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| ===Finite direct products===
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| It is possible to take the direct product of more than two groups at once. Given a finite sequence {{math|''G''<sub>1</sub>, ..., ''G''<sub>''n''</sub>}} of groups, the '''direct product'''
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| :<math>\prod_{i=1}^n G_i \;=\; G_1 \times G_2 \times \cdots \times G_n</math>
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| is defined as follows:
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| * The elements of {{math|''G''<sub>1</sub> × ··· × ''G''<sub>''n''</sub>}} are [[tuple]]s {{math|(''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>)}}, where {{math|''g<sub>i</sub>'' ∈ ''G<sub>i</sub>''}} for each {{math|''i''}}.
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| * The operation on {{math|''G''<sub>1</sub> × ··· × ''G''<sub>''n''</sub>}} is defined componentwise: <center>{{math|(''g''<sub>1</sub>, ..., ''g<sub>n</sub>'')(''g''<sub>1</sub>′, ..., ''g<sub>n</sub>''′)}} = {{math|(''g''<sub>1</sub>''g''<sub>1</sub>′, ..., ''g<sub>n</sub>g<sub>n</sub>''′)}}.</center>
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| This has many of the same properties as the direct product of two groups, and can be characterized algebraically in a similar way.
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| ===Infinite direct products===
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| It is also possible to take the direct product of an infinite number of groups. For an infinite sequence {{math|''G''<sub>1</sub>, ''G''<sub>2</sub>, ...}} of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples.
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| More generally, given an [[indexed family]] { {{math|''G<sub>i</sub>''}} }{{math|<sub>''i''∈''I''</sub>}} of groups, the '''direct product''' {{math|<big>∏</big><sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} is defined as follows:
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| * The elements of {{math|<big>∏</big><sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} are the elements of the [[Cartesian product#Infinite products|infinite Cartesian product]] of the sets {{math|''G<sub>i</sub>''}}, i.e. functions {{math|ƒ: ''I'' → }}<span style="font-size:170%;font-family:Sans-serif">U</span>{{math|<sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} with the property that {{math|ƒ(''i'') ∈ ''G<sub>i</sub>''}} for each {{math|''i''}}.
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| * The product of two elements {{math|ƒ, ''g''}} is defined componentwise: <center>{{math|(ƒ • ''g'')(''i'')}} = {{math|ƒ(''i'') • ''g''(''i'')}}.</center>
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| Unlike a finite direct product, the infinite direct product {{math|<big>∏</big><sub>''i''∈''I''</sub> ''G<sub>i</sub>''}} is not generated by the elements of the isomorphic subgroups { {{math|''G<sub>i</sub>''}} }{{math|<sub>''i''∈''I''</sub>}}. Instead, these subgroups generate a subgroup of the direct product known as the '''infinite direct sum''', which consists of all elements that have only finitely many non-identity components.
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| ===Other products===
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| ====Semidirect products====
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| {{main|Semidirect product}}
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| Recall that a group {{math|''P''}} with subgroups {{math|''G''}} and {{math|''H''}} is isomorphic to the direct product of {{math|''G''}} and {{math|''H''}} as long as it satisfies the following three conditions:
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| # The [[Intersection (set theory)|intersection]] {{math|''G'' ∩ ''H''}} is [[Trivial group|trivial]].
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| # Every element of {{math|''P''}} can be expressed as the product of an element of {{math|''G''}} and an element of {{math|''H''}}.
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| # Both {{math|''G''}} and {{math|''H''}} are [[normal subgroup|normal]] in {{math|''P''}}.
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| A '''semidirect product''' of {{math|''G''}} and {{math|''H''}} is obtained by relaxing the third condition, so that only one of the two subgroups {{math|''G'', ''H''}} is required to be normal. The resulting product still consists of ordered pairs {{math|(''g'', ''h'')}}, but with a slightly more complicated rule for multiplication.
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| It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group {{math|''P''}} is referred to as a '''[[Zappa–Szép product]]''' of {{math|''G''}} and {{math|''H''}}.
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| ====Free products====
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| {{main|Free product}}
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| The '''free product''' of {{math|''G''}} and {{math|''H''}}, usually denoted {{math|''G'' ∗ ''H''}}, is similar to the direct product, except that the subgroups {{math|''G''}} and {{math|''H''}} of {{math|''G'' ∗ ''H''}} are not required to commute. That is, if
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| :{{math|''G''}} = {{math|〈 ''S<sub>G</sub>'' }}<big>|</big>{{math| ''R<sub>G</sub>'' 〉}} and {{math|''H''}} = {{math|〈 ''S<sub>H</sub>'' }}<big>|</big>{{math| ''R<sub>H</sub>'' 〉}},
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| are presentations for {{math|''G''}} and {{math|''H''}}, then
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| :{{math|''G'' ∗ ''H''}} = {{math|〈 ''S<sub>G</sub>'' ∪ ''S<sub>H</sub>'' }}<big>|</big>{{math| ''R<sub>G</sub>'' ∪ ''R<sub>H</sub>'' 〉}}.
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| Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the [[coproduct]] in the [[category of groups]].
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| ====Subdirect products====
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| {{main|Subdirect product}}
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| If {{math|''G''}} and {{math|''H''}} are groups, a '''subdirect product''' of {{math|''G''}} and {{math|''H''}} is any subgroup of {{math|''G'' × ''H''}} which maps [[Surjective function|surjectively]] onto {{math|''G''}} and {{math|''H''}} under the projection homomorphisms. By [[Goursat's lemma]], every subdirect product is a fiber product, and vice versa.
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| ====Fiber products====
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| {{main|Pullback (category theory)}}
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| Let {{math|''G''}}, {{math|''H''}}, and {{math|''Q''}} be groups, and let {{math|''φ'': ''G'' → ''Q''}} and {{math|''χ'': ''H'' → ''Q''}} be [[epimorphism]]s. The '''fiber product''' of {{math|''G''}} and {{math|''H''}} over {{math|''Q''}}, also known as a '''pullback''', is the following subgroup of {{math|''G'' × ''H''}}:
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| : {{math|''G'' ×<sub>''Q''</sub> ''H''}} = <big>{</big> {{math|(''g'', ''h'') ∈ ''G'' × ''H'' : ''φ(g)'' {{=}} ''χ(h)''}} <big>}</big>.
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| By [[Goursat's lemma]], every subdirect product is a fiber product, and vice versa.
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| ==References==
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| <references />
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| * {{Citation
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| | last1=Artin
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| | first1=Michael
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| | authorlink1=Michael Artin
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| | title=Algebra
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| | publisher=[[Prentice Hall]]
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| | isbn=978-0-89871-510-1
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| | year=1991
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| }}
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| * {{Citation | last1=Herstein | first1=Israel Nathan |authorlink1 = Israel Nathan Herstein | title=Abstract algebra | publisher=Prentice Hall Inc. | location=Upper Saddle River, NJ | edition=3rd | isbn=978-0-13-374562-7 | mr=1375019 | year=1996}}.
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| * {{Citation | last1=Herstein | first1=Israel Nathan | title=Topics in algebra | publisher=Xerox College Publishing | location=Lexington, Mass. | edition=2nd | mr=0356988 | year=1975}}.
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| * {{Lang Algebra}}<!-- Don't add a fullstop here: it breaks the layout! -->
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| * {{Citation | last1=Lang | first1=Serge | title=Undergraduate Algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=3rd | isbn=978-0-387-22025-3 | year=2005}}.
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| * {{Citation | last1=Robinson | first1=Derek John Scott | title=A course in the theory of groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996}}.
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| [[Category:Group theory]]
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| [[ca:Producte directe#Producte directe de grups]]
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| [[de:Direktes Produkt#Direktes Produkt von Gruppen]]
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| [[es:Producto directo]]
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| [[it:Prodotto diretto]]
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| [[nl:Direct product]]
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| [[pl:Iloczyny grup]]
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| [[ru:Прямое произведение#Прямое произведение групп]]
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| [[zh:直积]]
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